行列の計算例題(行列式(余因子展開))

          


例題:次の行列式を余因子展開を使って計算せよ。

\[ \left|\,\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}@{}}0 & 2 & 0 & 0 \\ 0 & 2 & 0 & 2 \\ {-}1 & 2 & {-}2 & {-}1 \\ 2 & 0 & 0 & 0\end{array}\,\right| \]

解答

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}0 & 2 & 0 & 0 \\ 0 & 2 & 0 & 2 \\ {-}1 & 2 & {-}2 & {-}1 \\ 2 & 0 & 0 & 0\end{array}\,\right| \qquad \) (余因子展開) 第4列で展開する : [0, 2, -1, 0]

\( \qquad\quad = (-1)^{4-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & 2 & 0 \\ {-}1 & 2 & {-}2 \\ 2 & 0 & 0\end{array}\,\right| + (-1)^{4-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & 2 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-2}\times2\times(-8) + (-1)^{4-3}\times(-1)\times0 \)

\( \qquad\quad = -16 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & 2 & 0 \\ {-}1 & 2 & {-}2 \\ 2 & 0 & 0\end{array}\,\right| \qquad \) (余因子展開) 第3列で展開する : [0, -2, 0]

\( \qquad\quad = (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 2 \\ 2 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-2}\times(-2)\times(-4) \)

\( \qquad\quad = -8 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 2 \\ 2 & 0\end{array}\,\right| = 0\times0 - 2\times2 = -4 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & 2 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 0\end{array}\,\right| \qquad \) (余因子展開) 第3列で展開する : [0, 0, 0]

\( \qquad\quad = \)

\( \qquad\quad = \)

\( \qquad\quad = 0 \)